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<DIV>It's not just those values. You want all ratios up to a certain "size" to be approximated well - increasing this size is more important that getting a better approximation. There is no obvious definition for size. In general, divisibility only by small primes is important; 9/8 is clearly superior to 7/6, for example.</DIV>
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<DIV>31 works pretty well.</DIV>
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<DIV>Franklin T. Adams-Watters<BR>16 W. Michigan Ave.<BR>Palatine, IL 60067<BR>847-776-7645</DIV>
<DIV> </DIV> <BR>-----Original Message-----<BR>From: Jud McCranie <j.mccranie@adelphia.net><BR>To: David Wilson <davidwwilson@comcast.net><BR>Cc: Sequence Fans <seqfan@ext.jussieu.fr><BR>Sent: Wed, 18 Jan 2006 13:46:40 -0500<BR>Subject: Re: Musical sequence<BR><BR>
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<DIV class=AOLPlainTextBody id=AOLMsgPart_0_f038a4a1-6e5e-4093-b035-5feca539578d>At 01:40 PM 1/18/2006, David Wilson wrote: <BR>>In general music theory, we hold to the concept of an octave, >specifically, that if a musical scale includes a tone of frequency f, it >also includes a tone of frequency 2f. <BR>> <BR>>What if we abandon the notion of octave, but keep the idea of equally >spaced tones, so that our scale can be described by base tone frequency b >and a ratio r between adjacent tones on the scale. <BR>> <BR>>Is there a measure of goodness of r that would rate r = 2^(1/12) high? >Presumably this rating would be based on the closeness of tone ratios to >simple rationals which represent pleasing harmonies. Given this rating, >what would be the best r? If we return to octave-based scales, could be >construct a sequence of t with increasingly better ratings of 2^(1/t), >which would represent increasingly !
good choices for number of tones in an >octave? <BR> <BR>I looked at this sometime ago. The important thing about 12 equally-spaced tones is that r^4 is close to 5/4, r^5 is close to 4/3, and r^7 is close to 3/2, r is the 12-th root of 2, as above. The next values that come close to these important values are when there are 43 and 53 equally-spaced tones between frequencies f and 2f, IIRC. <BR> <BR></DIV><!-- end of AOLMsgPart_0_f038a4a1-6e5e-4093-b035-5feca539578d --></DIV></DIV>
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